Method and system for fast detection of lines in medical images

ABSTRACT

A method an apparatus for detecting lines in medical images is disclosed, wherein a direction image array and a line image array are formed by filtering a digital image with a single-peaked filter, convolving the resultant array with second order difference operators oriented along the horizontal, vertical, and diagonal axes, and computing the direction image arrays and line image arrays as direct scalar functions of the results of the second order difference operations. Advantageously, line detection based on the use of four line operator functions along the horizontal, vertical, and diagonal directions in accordance with the preferred embodiments actually results in fewer computations than line detection based on the use of three line operator functions. In particular, because of the special symmetries involved, 3×3 second order difference operators may be effectively used. Moreover, the number of computations associated with the second order difference operations may be achieved with simple register shifts, additions, and subtractions, yielding an overall line detection process that is significantly less computationally intensive than prior art algorithms. Also according to a preferred embodiment, computational complexity is reduced by selecting a separable single-peaked filter, and sequentially convolving the digital image with the component kernels of the separable single-peaked filter.

FIELD OF THE INVENTION

[0001] The present invention relates to the field of computer aidedanalysis of medical images. In particular, the present invention relatesto a fast method for detecting lines in medical images.

BACKGROUND OF THE INVENTION

[0002] Line detection is an important first step in many medical imageprocessing algorithms. For example, line detection is an important earlystep of the algorithm disclosed in U.S. patent application Ser. No.08/676,660, entitled “Method and Apparatus for Fast Detection ofSpiculated Lesions in Digital Mammograms,” filed Jul. 19, 1996, thecontents of which are hereby incorporated by reference into the presentapplication. Generally speaking, if the execution time of the linedetection step can be shortened, then the execution time of the overallmedical image processing algorithm employing that line detection stepcan be shortened.

[0003] In order to clearly illustrate the features and advantages of thepreferred embodiments, the present disclosure will describe the linedetection algorithms of both the prior art and the preferred embodimentsin the context of the computer-assisted diagnosis system of U.S. patentapplication Ser. No. 08/676,660, supra. Importantly, however, the scopeof the preferred embodiments is not so limited, the features andadvantages of the preferred embodiments being applicable to a variety ofimage processing applications.

[0004]FIG. 1 shows steps performed by a computer-assisted diagnosis unitsimilar to that described in U.S. patent application Ser. No.08/676,660, which is adapted to detect abnormal spiculations or lesionsin digital mammograms. At step 102, an x-ray mammogram is scanned in anddigitized into a digital mammogram. The digital mammogram may be, forexample, a 4000×5000 array of 12-bit gray scale pixel values. Such adigital mammogram would generally correspond to a typical 8″×10″ x-raymammogram which has been digitized at 50 microns (0.05 mm) per pixel.

[0005] At step 104, which is generally an optional step, the digitalmammogram image is locally averaged, using steps known in the art, downto a smaller size corresponding, for example, to a 200 micron (0.2 mm)spatial resolution. The resulting digital mammogram image that isprocessed by subsequent steps is thus approximately 1000×1250 pixels. Asis known in the art, a digital mammogram may be processed at differentresolutions depending on the type of features being detected. If, forexample, the scale of interest is near the order of magnitude 1 mm-10mm, i.e., if lines on the order of 1 mm-10 mm are being detected, it isneither efficient nor necessary to process a full 50-micron (0.05 mm)resolution digital mammogram. Instead, the digital mammogram isprocessed at a lesser resolution such as 200 microns (0.2 mm) per pixel.

[0006] Generally speaking, it is to be appreciated that the advantagesand features of the preferred embodiments disclosed infra are applicableindependent of the size and spatial resolution of the digital mammogramimage that is processed. Nevertheless, for clarity of disclosure, andwithout limiting the scope of the preferred embodiments, the digitalmammogram images in the present disclosure, which will be denoted by thesymbol I, will be M×N arrays of 12-bit gray scale pixel values, with Mand N having exemplary values of 1000 and 1250, respectively.

[0007] At step 106, line and direction detection is performed on thedigital mammogram image I. At this step, an M×N line image L(i, j) andan M×N direction image θ_(max)(i, j) are generated from the digitalmammogram image I. The M×N line image L(i, j) generated at step 106comprises, for each pixel (i, j), line information in the form of a “1”if that pixel has a line passing through it, and a “0” otherwise. TheM×N direction image θ_(max)(i, j) comprises, for those pixels (i, j)having a line image value of “1”, the estimated direction of the tangentto the line passing through the pixel (i, j). Alternatively, of course,the direction image θ_(max)(i, j) may be adjusted by 90 degrees tocorrespond to the direction orthogonal to the line passing through thepixel (i, j).

[0008] At step 108, information in the line and direction images isprocessed for determining the locations and relative priority ofspiculations in the digital mammogram image I. The early detection ofspiculated lesions (“spiculations”) in mammograms is of particularimportance because a spiculated breast tumor has a relatively highprobability of being malignant.

[0009] Finally, at step 110, the locations and relative priorities ofsuspicious spiculated lesions are output to a display device for viewingby a radiologist, thus drawing his or her attention to those areas. Theradiologist may then closely examine the corresponding locations on theactual film x-ray mammogram. In this manner, the possibility of misseddiagnosis due to human error is reduced.

[0010] One of the desired characteristics of a spiculation-detecting CADsystem is high speed to allow processing of more x-ray mammograms inless time. As indicated by the steps of FIG. 1, if the execution time ofthe line and direction detection step 106 can be shortened, then theexecution time of the overall mammogram spiculation detection algorithmcan be shortened.

[0011] A first prior art method for generating line and direction imagesis generally disclosed in Gonzales and Wintz, Digital Image Processing(1987) at 333-34. This approach uses banks of filters, each filter being“tuned” to detect lines in a certain direction. Generally speaking, this“tuning” is achieved by making each filter kernel resemble asecond-order directional derivative operator in that direction. Eachfilter kernel is separately convolved with the digital mammogram imageI. Then, at each pixel (i, j), line orientation can be estimated byselecting the filter having the highest output at (i, j), and linemagnitude may be estimated from that output and other filter outputs.The method can be generalized to lines having pixel widths greater than1 in a multiscale representation shown in Daugman, “Complete Discrete2-D Gabor Transforms by Neural Networks for Image Analysis andCompression,” IEEE Trans. ASSP, Vol. 36, pp. 1169-79 (1988).

[0012] The above filter-bank algorithms are computationally intensive,generally requiring a separate convolution operation for eachorientation-selective filter in the filter bank. Additionally, theaccuracy of the angle estimate depends on the number of filters in thefilter bank, and thus there is an implicit tradeoff between the size ofthe filter bank (and thus total computational cost) and the accuracy ofangle estimation.

[0013] A second prior art method of generating line and direction imagesis described in Karssemeijer, “Recognition of Stellate Lesions inDigital Mammograms,” Digital Mammography: Proceedings of the 2ndInternational Workshop on Digital Mammography, York, England, (Jul.10-12, 1994) at 211-19, and in Karssemeijer, “Detection of StellateDistortions in Mammograms using Scale Space Operators,” InformationProcessing in Medical Imaging 335-46 (Bizais et al., eds. 1995) at335-46. A mathematical foundation for the Karssemeijer approach is foundin Koenderink and van Doorn, “Generic Neighborhood Operators,” IEEETransactions on Pattern Analysis and Machine Intelligence, Vol. 14, No.6 (June 1992) at 597-605. The contents of each of the above twoKarssemeijer references and the above Koenderink reference are herebyincorporated by reference into the present application.

[0014] The Karssemeijer algorithm uses scale space theory to provide anaccurate and more efficient method of line detection relative to thefilter-bank method. More precisely, at a given level of spatial scale σ,Karssemeijer requires the convolution of only three kernels with thedigital mammogram image I, the angle estimation at a pixel (i, j) thenbeing derived as a trigonometric function of the three convolutionresults at (i, j).

[0015]FIG. 2 shows steps for computing line and direction images inaccordance with the Karssemeijer algorithm. At step 202, a spatial scaleparameter a and a filter kernel size N_(k) are selected. The spatialscale parameter o dictates the width, in pixels, of a Gaussian kernelG(r,σ), the equation for which is shown in Eq. (1):

G(r,σ)=(½πσ²)exp(−r ²/2σ²)  (1)

[0016] At step 202, the filter kernel size N_(k), in pixels, isgenerally chosen to be large enough to contain the Gaussian kernelG(r,σ) in digital matrix form, it being understood that the functionG(r,σ) becomes quite small very quickly. Generally speaking, the spatialscale parameter σ corresponds, in an order-of-magnitude sense, to thesize of the lines being detected. By way of example only, and not by wayof limitation, for detecting 1 mm-10 mm lines in fibrous breast tissuein a 1000×1250 digital mammogram at 200 micron (0.2 mm) resolution, thevalue of σ may be selected as 1.5 pixels and the filter kernel sizeN_(k) may be selected as 11 pixels. For detecting different size linesor for greater certainty of results, the algorithm or portions thereofmay be repeated using different values for a and the kernel size.

[0017] At step 204, three filter kernels K_(σ)(0), K_(σ)(60), andK_(σ)(120) are formed as the second order directional derivatives of theGaussian kernel G(r,σ) at 0 degrees, 60 degrees, and 120 degrees,respectively. The three filter kernels K_(σ)(0), K_(σ)(60), andK_(σ)(120) are each of size N_(k), each filter kernel thus containingN_(k×)N_(k) elements.

[0018] At step 206, the digital mammogram image I is separatelyconvolved with each of the three filter kernels K_(σ)(0), K_(σ)(60), andK(120) to produce three line operator functions W_(σ)(0), W_(σ)(60), andW_(σ)(120), respectively, as shown in Eq. (2):

W _(σ)(0)=I*K _(σ)(0)W _(σ)(60)=I*K _(σ)(60)W _(σ)(120)=I*K_(σ)(120)  (2)

[0019] Each of the line operator functions W_(σ)(0), W_(σ)(60), andW_(σ)(120) is, of course, a two-dimension array that is slightly largerthan the original M×N digital mammogram image array I due to the sizeN_(k) of the filter kernels.

[0020] Subsequent steps of the Karssemeijer algorithm are based on arelation shown in Koenderink, supra, which shows that an estimationfunction W_(σ)(θ) may be formed as a combination of the line operatorfunctions W_(σ)(0), W_(σ)(60), and W_(σ)(120) as defined in equation(3):

W _(σ)(θ)=(⅓)(1+2 cos(2θ))W _(σ)(0)+(⅓)(1− cos(2θ)+({squareroot}3)sin(2θ))W _(σ)(60)+(⅓)(1− cos(2θ)−({squareroot}3)sin(2θ))W_(σ)(120)  (3)

[0021] As indicated by the above definition, the estimation functionW_(σ)(θ) is a function of three variables, the first two variables beingpixel coordinates (i, j) and the third variable being an angle θ. Foreach pixel location (i, j), the estimation function W_(σ)(θ) representsa measurement of line strength at pixel (i, j) in the directionperpendicular to θ. According to the Karssemeijer method, an analyticalexpression for the extrema of W_(σ)(θ) with respect to θ, denotedθ_(min,max) at a given pixel (i, j) is given by Eq. (4):

θ_(min,max)=½[arc tan{({square root}3)(W _(σ)(60)−W ₉₄ (120))/(W_(σ)(60)+W _(σ)(120)−2W _(σ)(0))}±π]  (4)

[0022] Thus, at step 208, the expression of Eq. (4) is computed for eachpixel based on the values of W_(σ)(0), W_(σ)(60), and W_(σ)(120) thatwere computed at step 206. Of the two solutions to equation (4), thedirection θ_(max) is then selected as the solution that yields thelarger magnitude for W_(σ)(θ) at that pixel, denoted W_(σ)(θ_(max)).Thus, at step 208, an array θ_(max)(i, j) is formed that constitutes thedirection image corresponding to the digital mammogram image I. As anoutcome of this process, a corresponding two-dimensional array of lineintensities corresponding to the maximum direction θ_(max) at each pixelis formed, denoted as the line intensity function W_(σ)(θ_(max)).

[0023] At step 210, a line image L(i, j) is formed using informationderived from the line intensity function W_(σ)(θ_(max)) that wasinherently generated during step 208. The array L(i, j) is formed fromW_(σ)(θ_(max)) using known methods such as a simple thresholding processor a modified thresholding process based on a histogram ofW_(σ)(θ_(max)). With the completion of the line image array L(i, j) andthe direction image array θ_(max)(i, j), the line detection process iscomplete.

[0024] Optionally, in the Karssemeijer algorithm a plurality of spatialscale values σ1, σ2, . . . , σn may be selected at step 202. The steps204-210 are then separately carried out for each of the spatial scalevalues (σ1, σ2, . . . , σn. For a given pixel (i, j), the value ofθ_(max)(i, j) is selected to correspond to the largest value amongW_(σ1)(θ_(max1)), W_(σ2)(θ_(max2)), . . . , W_(σn)(θ_(maxn)). The lineimage L(i, j) is formed by thresholding an array corresponding tolargest value among W_(σ1)(θ_(max1)), W_(σ2)(θ_(max2)), . . . ,W_(σn)(θ_(maxn)) at each pixel.

[0025] Although it is generally more computationally efficient than thefilter-bank method, the prior art Karssemeijer algorithm hascomputational disadvantages. In particular, for a given spatial scaleparameter σ, the Karssemeijer algorithm requires three separateconvolutions of N_(k)×N_(k) kernels with the M×N digital mammogram imageI. Each convolution, in turn, requires approximately M·N·(N_(k))²multiplication and addition operations, which becomes computationallyexpensive as the kernel size N_(k), which is proportional to the spatialscale parameter σ, grows. Thus, for a constant digital mammogram imagesize, the computational intensity of the Karssemeijer algorithmgenerally grows according to the square of the scale of interest.

[0026] Accordingly, it would be desirable to provide a line detectionalgorithm for use in a medical imaging system that is lesscomputationally intensive, and therefore faster, than the above priorart algorithms.

[0027] It would further be desirable to provide a line detectionalgorithm for use in a medical imaging system that is capable ofoperating at multiple spatial scales for detecting lines of varyingwidths.

[0028] It would be even further desirable to provide a line detectionalgorithm for use in a medical imaging system in which, as the scale ofinterest grows, the computational intensity grows at a rate less thanthe rate of growth of the square of the scale of interest.

SUMMARY OF THE INVENTION

[0029] These and other objects are provided for by a-method andapparatus for detecting lines in a medical imaging system by filteringthe digital image with a single-peaked filter, convolving the resultantarray with second order difference operators oriented along thehorizontal, vertical, and diagonal axes, and computing direction imagearrays and line image arrays as direct scalar functions of the resultsof the second order difference operations. Advantageously, it has beenfound that line detection based on the use of four line operatorfunctions can actually require fewer computations than line detectionbased on the use of three line operator functions, if the four lineoperator functions correspond to the special orientations of 0, 45, 90,and 135 degrees. Stated another way, it has been found that the numberof required computations is significantly reduced where the aspect ratioof the second order difference operators corresponds to the angulardistribution of the line operator functions. Thus, where the secondorder difference operators are square kernels, having an aspect ratio ofunity, the preferred directions of four line operator functions is at 0,45, 90, and 135 degrees.

[0030] In a preferred embodiment, a spatial scale parameter is selectedthat corresponds to a desired range of line widths for detection. Thedigital image is then filtered with a single-peaked filter having a sizerelated to the spatial scale parameter, to produce a filtered imagearray. The filtered image array is separately convolved with secondorder difference operators at 0, 45, 90, and 135 degrees. The directionimage array and the line image array are then computed at each pixel asscalar functions of the elements of the arrays resulting from theseconvolutions. Because of the special symmetries involved, the secondorder difference operators may be 3×3 kernels. Moreover, the number ofcomputations associated with the second order difference operations maybe achieved with simple register shifts, additions, and subtractions,yielding an overall line detection process that is significantly lesscomputationally intensive than prior art algorithms.

[0031] In another preferred embodiment, the digital image is firstconvolved with a separable single-peaked filter kernel, such as aGaussian. Because a separable function may be expressed as theconvolution of a first one dimensional kernel and a second onedimensional kernel, the convolution with the separable single-peakedfilter kernel is achieved by successive convolutions with a first onedimensional kernel and a second one dimensional kernel, whichsignificantly reduces computation time in generating the filtered imagearray. The filtered image array is then convolved with three 3×3 secondorder difference operators, the first such operator comprising thedifference between a horizontal second order difference operator and avertical difference operator, the second such operator comprising thedifference between a first diagonal second order difference operator anda second diagonal second order difference operator, and the third suchoperator being a Laplacian operator. Because of the special symmetriesassociated with the selection of line operator functions at 0, 45, 90,and 135 degrees, the direction image array and the line image array arethen computed at each pixel as even simpler scalar functions of theelements of the arrays resulting from the three convolutions.

[0032] Thus, line detection algorithms in accordance with the preferredembodiments are capable of generating line and direction images usingsignificantly fewer computations than prior art algorithms by takingadvantage of the separability of Gaussians and other symmetric filterkernels, while also taking advantage of discovered computationalsimplifications that result from the consideration of four line operatorfunctions oriented in the horizontal, vertical, and diagonal directions.

BRIEF DESCRIPTION OF THE DRAWINGS

[0033]FIG. 1 shows steps taken by a computer-aided diagnosis (“CAD”)system for detecting spiculations in digital mammograms in accordancewith the prior art.

[0034]FIG. 2 shows line detection steps taken by the CAD system of FIG.1.

[0035]FIG. 3 shows line detection steps according to a preferredembodiment.

[0036]FIG. 4 shows steps for convolution with second order directionalderivative operators in accordance with a preferred embodiment.

[0037]FIG. 5 shows line detection steps according to another preferredembodiment.

DETAILED DESCRIPTION

[0038]FIG. 3 shows steps of a line detection algorithm in accordancewith a preferred embodiment. At step 302, a spatial scale parameter θand a filter kernel size N_(k) are selected in manner similar to that ofstep 202 of FIG. 2. However, in a line detection system according to apreferred embodiment, it is possible to make these factors larger thanwith the prior art system of FIG. 2 while not increasing thecomputational intensity of the algorithm. Alternatively, in a linedetection system according to a preferred embodiment, these factors mayremain the same as with the prior art system of FIG. 2 and thecomputational intensity of the algorithm will be reduced. As a furtheralternative, in a line detection system according to a preferredembodiment, it is possible to detect lines using a greater number ofdifferent spatial scales of interest σ while not increasing thecomputational intensity of the algorithm.

[0039] At step 304, the digital mammogram image I is convolved with atwo-dimensional single-peaked filter F having dimensions N_(k)×N_(k) toform a filtered image array I_(F) as shown in Eq. (5):

I _(F) =I*F  (5)

[0040] By single-peaked filter, it is meant that the filter F is afunction with a single maximum point or single maximum region. Examplesof such a filter include the Gaussian, but may also include other filterkernels such as a Butterworth filter, an inverted triangle or parabola,or a flat “pillbox” function. It has been found, however, that aGaussian filter is, the most preferable. The size of the single-peakedfilter F is dictated by the spatial scale parameter σ. For example,where a Gaussian filter is used, σ is the standard deviation of theGaussian, and where a flat pillbox function is used, σ corresponds tothe radius of the pillbox. In subsequent steps it is assumed that aGaussian filter is used, although the algorithm may be adapted by oneskilled in the art to use other filters.

[0041] At step 306, the filtered image array I_(F) is then separatelyconvolved with second order directional derivative operators. Inaccordance with a preferred embodiment, it is computationallyadvantageous to compute four directional derivatives at 0, 45, 90, and135 degrees by convolving filtered image array I_(F) with second orderdirectional derivative operators D₂(0), D₂(45), D₂(90), and D₂(135) toproduce the line operator functions W_(σ)(0), W_(σ)(45), W_(σ)(90), andW_(σ)(135), respectively, as shown in Eqs. (6a)-(6d).

W _(σ)(0)=I _(F) *D ₂(0)  (6a)

W _(σ)(45)=I _(F) *D ₂(45)  (6b)

W _(σ)(90)=I _(F) *D ₂(90)  (6c)

W _(σ)(135)=I _(F) *D ₂(135)  (6d)

[0042] Advantageously, because the particular directions of 0, 45, 90,and 135 degrees are chosen, these directional derivative operators arepermitted to consist of the small 3×3 kernels shown in Eqs. (7a)-(7d):$\begin{matrix}\begin{matrix}{0\quad} & 0 & 0 & \quad & \quad \\{D_{2}(0)} & = & {- 1} & 2 & {- 1} \\{0\quad} & 0 & 0 & \quad & \quad\end{matrix} & \text{(7a)} \\\begin{matrix}{0\quad} & 0 & {- 1} & \quad & \quad \\{D_{2}(45)} & = & 0 & 2 & 0 \\{{- 1}\quad} & 0 & 0 & \quad & \quad\end{matrix} & \text{(7b)} \\\begin{matrix}{0\quad} & {- 1} & 0 & \quad & \quad \\{D_{2}(90)} & = & 0 & 2 & 0 \\{0\quad} & {- 1} & 0 & \quad & \quad\end{matrix} & \text{(7c)} \\\begin{matrix}{{- 1}\quad} & 0 & 0 & \quad & \quad \\{D_{2}(135)} & = & 0 & 2 & 0 \\{0\quad} & 0 & {- 1} & \quad & \quad\end{matrix} & \text{(7d)}\end{matrix}$

[0043] The above 3×3 second order directional derivative operators arepreferred, as they result in fewer computations than larger second orderdirectional derivative operators while still providing a good estimateof the second order directional derivative when convolved with thefiltered image array I_(F). However, the scope of the preferredembodiments is not necessarily so limited, it being understood thatlarger operators for estimating the second order directional derivativesmay be used if a larger number of computations is determined to beacceptable. For a minimal number of computations in accordance with apreferred embodiment, however, 3×3 kernels are used.

[0044] Subsequent steps are based on an estimation function W_(σ)(θ)that can be formed from the arrays W_(σ)(0), W_(σ)(45), W_(σ)(90), andW_(σ)(135) by adapting the formulas in Koenderink, supra, for fourestimators spaced at intervals of 45 degrees. The resulting formula isshown below in Eq. (8).

W _(σ)(θ)=¼{(1+2 cos(2θ))W _(σ)(0)+(1+2 sin(2θ))W _(σ)(45)+(1−2cos(2θ))W _(σ)(90)+(1−2 sin(2θ))W _(σ)(135)}  (8)

[0045] It has been found that the extrema of the estimation functionW_(σ)(θ) with respect to θ, denoted θ_(min,max) at a given pixel (i, j)is given by Eq. (9):

θ _(min,max)=½[a tan{(W _(σ)(45)−W _(σ)(135))/(W _(σ)(0)−W_(σ)(90))}±π]  (9)

[0046] At step 308, the expression of Eq. (9) is computed for eachpixel. Of the two solutions to equation (4), the direction θ_(max) isthen selected as the solution that yields the larger magnitude forW_(σ)(θ) at that pixel, denoted as the line intensity W_(σ)(θ_(max)).Thus, at step 308, an array θ_(max)(i, j) is formed that constitutes thedirection image corresponding to the digital mammogram image I. As anoutcome of this process, a corresponding two-dimensional array of lineintensities corresponding to the maximum direction θ_(max) at each pixelis formed, denoted as the line intensity function W_(σ)(θ_(max)).

[0047] At step 310, a line image array L(i, j) is formed usinginformation derived from the line intensity function W_(σ)(θ_(max)) thatwas inherently generated during step 308. The line image array L(i, j)is formed from the line intensity function W_(σ)(θ_(max)) using knownmethods such as a simple thresholding process or a modified thresholdingprocess based on a histogram of the line intensity functionW_(σ)(θ_(max)). With the completion of the line image array L(i, j) andthe direction image array θ_(max)(i, j), the line detection process iscomplete.

[0048]FIG. 4 illustrates unique computational steps corresponding to thestep 306 of FIG. 3. At step 306, the filtered image array I_(F) isconvolved with the second order directional derivative operators D₂(0),D₂(45), D₂(90), and D₂(135) shown in Eq. (7). An advantage of the use ofthe small 3×3 kernels D₂(0), D₂(45), D₂(90), and D₂(135) evidencesitself in the convolution operations corresponding to step 306. Inparticular, because each of the directional derivative operators hasonly 3 nonzero elements −1, 2, and −1, general multiplies are notnecessary at all in step 306, as the multiplication by 2 justcorresponds to a single left bitwise register shift and themultiplications by −1 are simply sign inversions. Indeed, eachconvolution operation of Eq. (6) can be simply carried out at each pixelby a single bitwise left register shift followed by two subtractions ofneighboring pixel values from the shifted result.

[0049] Thus, at step 402 each pixel in the filtered image array I_(F) isdoubled to produce the doubled filtered image array 2I_(F). This can beachieved through a multiplication by 2 or, as discussed above, a singlebitwise left register shift. At step 404, at each pixel (i, j) in thearray 2I_(F), the value of I_(F)(i−1,j) is subtracted, and at step 406,the value of I_(F)(i+1,j) is subtracted, the result being equal to thedesired convolution result I_(F)*D₂(0) at pixel (i, j). Similarly, atstep 408, at each pixel (i, j) in the array 2I_(F), the value ofI_(F)(i−1,j−1) is subtracted, and at step 410, the value ofI_(F)(i+1,j+1) is subtracted, the result being equal to the desiredconvolution result I_(F)*D₂(45) at pixel (i, j). Similarly, at step 412,at each pixel (i, j) in the array 2I_(F), the value of I_(F)(i, j−1) issubtracted, and at step 414, the value of I_(F)(i, j+1) is subtracted,the result being equal to the desired convolution result I_(F)*D₂(90) atpixel (i, j). Finally, at step 416, at each pixel (i, j) in the array2I_(F), the value of I_(F)(i+1,j−1) is subtracted, and at step 418, thevalue of I_(F)(i−1,j+1) is subtracted, the result being equal to thedesired convolution result I_(F)*D₂(135) at pixel (i, j). The steps406-418 are preferably carried out in the parallel fashion shown in FIG.4 but can generally be carried out in any order.

[0050] Thus, it is to be appreciated that in the embodiment of FIGS. 3and 4 a line detection algorithm is executed using four line operatorfunctions W_(σ)(0), W_(σ)(45), W_(σ)(90), and W_(σ)(135) while at thesame time using fewer computations than the Karssemeijer algorithm ofFIG. 2, which uses only three line operator functions W_(σ)(0),W_(σ)(60), W_(σ)(120). In accordance with a preferred embodiment, thealgorithm of FIGS. 3 and 4 takes advantage of the interchangeability ofthe derivative and convolution operations while also taking advantage ofthe finding that second order directional derivative operators in eachof the four directions 0, 45, 90, and 135 degrees may be implementedusing small 3×3 kernels each having only three nonzero elements −1, 2,and −1. In the Karssemeijer algorithm of FIG. 2, there are threeconvolutions of the M×N digital mammogram image I with the N_(k)×N_(k)kernels, requiring approximately 3·(N_(k))²·M·N multiplications and addsto derive the three line estimator functions W_(σ)(0), W_(σ)(60), andW_(σ)(120). However, in the embodiment of FIGS. 3 and 4, the computationof the four line estimator functions W₉₄ (0), W_(σ)(45), W_(σ)(90), andW_(σ)(135) requires a first convolution requiring (N_(k))²·M·Nmultiplications, followed by M·N doubling operations and 8·M·Nsubtractions, which is a very significant computational advantage. Theremaining portions of the different algorithms take approximately thesame amount of computations once the line estimator functions arecomputed.

[0051] For illustrative purposes in comparing the algorithm of FIGS. 3and 4 with the prior art Karssemeijer algorithm of FIG. 2, let us assumethat the operations of addition, subtraction, and register-shiftingoperation take 10 clock cycles each, while the process of multiplicationtakes 30 clock cycles. Let us further assume that an exemplary digitalmammogram of M×N=1000×1250 is used and that N_(k) is 11. For comparisonpurposes, it is most useful to look at the operations associated withthe required convolutions, as they require the majority of computationaltime. For this set of parameters, the Karssemeijer algorithm wouldrequire 3(11)²(1000)(1250)(30+10)=18.2 billion clock cycles to computethe three line estimator functions W_(σ)(0), W_(σ)(60), and W_(σ)(120).In contrast, the algorithm of FIGS. 3 and 4 would require only(11)²(1000)(1250)(30+10)+(1250)(1000)(10)+8(1250)(1000)(10)=6.2 billionclock cycles to generate the four line operator functions W_(σ)(0),W_(σ)(45), W_(σ)(90), and W_(σ)(135), a significant computationaladvantage.

[0052]FIG. 5 shows steps of a line detection algorithm in accordancewith another preferred embodiment. It has been found that the algorithmof FIGS. 3 and 4 can be made even more computationally efficient wherethe single-peaked filter kernel F is selected to be separable. Generallyspeaking, a separable kernel can be expressed as a convolution of twokernels of lesser dimensions, such as one-dimensional kernels. Thus, theN_(k)×N_(k) filter kernel F(i, j) is separable where it can be formed asa convolution of an N_(k)×1 kernel F_(x)(i) and a 1×N_(k) kernelF_(y)(j), i.e., F(i, j)=F_(x)(i)*F_(y)(j). As known in the art, anN_(k)×1 kernel is analogous to a row vector of length N_(k) while a1×N_(k) kernel is analogous to a column vector of length N_(k).

[0053] Although a variety of single-peaked functions are within thescope of the preferred embodiments, the most optimal function has beenfound to be the Gaussian function of Eq. (1), supra. For purposes of theembodiment of FIG. 5, and without limiting the scope of the preferredembodiments, the filter kernel notation F will be replaced by thenotation G to indicate that a Gaussian filter is being used:$\begin{matrix}\begin{matrix}{G = {\left( {{1/2}{\pi\sigma}^{2}} \right){\exp \left( {{{- x^{2}}/2}\sigma^{2}} \right)}{\exp \left( {{{- y^{2}}/2}\sigma^{2}} \right)}}} \\{= {G_{x}*G_{y}}}\end{matrix} & (10) \\\begin{matrix}{G_{x} = \quad \left\lbrack {g_{x,0}\quad g_{x,1}\quad g_{x,2}\quad \cdots \quad g_{x,{{Nk} - 1}}} \right\rbrack} \\{\quad g_{y,0}} \\{\quad g_{y,1}}\end{matrix} & (11) \\\begin{matrix}{G_{y} = \quad g_{y,3}} \\{\quad \vdots} \\{\quad g_{y,{{Nk} - 1}}}\end{matrix} & (12)\end{matrix}$

[0054] At step 502, the parameters σ and N_(k) are selected in a mannersimilar to step 302 of FIG. 3. It is preferable for N_(k) to be selectedas an odd number, so that a one-dimensional Gaussian kernel of lengthN_(k) may be symmetric about its central element. At step 504, the M×Ndigital mammogram image I is convolved with the Gaussian N_(k)×1 kernelG_(x) to produce an intermediate array I_(x):

I _(x) =G _(x) *I  (13)

[0055] In accordance with a preferred embodiment, the sigma of theone-dimensional Gaussian kernel G_(x) is the spatial scale parameter aselected at step 502. The intermediate array I_(x) resulting from step504 is a two-dimensional array having dimensions of approximately(M+2N_(k))×N.

[0056] At step 506, the intermediate array I_(x) is convolved with theGaussian 1×N_(k) kernel G_(y) to produce a Gaussian-filtered image arrayI_(G):

I _(G) =I _(x) *G _(y)  (14)

[0057] In accordance with a preferred embodiment, the sigma of theone-dimensional Gaussian kernel G_(y) is also the spatial scaleparameter a selected at step 502. The filtered image array I_(G)resulting from step 506 is a two-dimensional array having dimensions ofapproximately (M+2N_(k))×(N+2N_(k)). Advantageously, because of theseparability property of the two-dimensional Gaussian, the filteredimage array I_(G) resulting from step 506 is identical to the result ofa complete two-dimensional convolution of an N_(k)×N_(k) Gaussian kerneland the digital mammogram image I. However, the number ofmultiplications and additions is reduced to 2·N_(k)·M·N instead of(N_(k))²·M·N.

[0058] Even more advantageously, in the situation where N_(k) isselected to be an odd number and the one-dimensional Gaussian kernelsare therefore symmetric about a central element, the number ofmultiplications is reduced even further. This computational reductioncan be achieved because, if N_(k) is odd, then the component onedimensional kernels G_(x) and G_(y) are each symmetric about a centralpeak element. Because of this relation, the image values correspondingto symmetric kernel locations can be added prior to multiplication bythose kernel values, thereby reducing by half the number of requiredmultiplications during the computations of Eqs. (13) and (14).Accordingly, in a preferred embodiment in which N_(k) is selected to bean odd number, the number of multiplications associated with therequired convolutions is approximately N_(k)·M·N and the number ofadditions is approximately 2·N_(k)·M·N.

[0059] In addition to the computational savings over the embodiment ofFIGS. 3 and 4 due to filter separability, it has also been found thatthe algorithm of FIGS. 3 and 4 may be made even more efficient by takingadvantage of the special symmetry of the spatial derivative operators at0, 45, 90, and 135 in performing operations corresponding to steps306-310. In particular, it has been found that for each pixel (i, j),the solution for the direction image θ_(max) and the line intensityfunction W_(σ)(θ_(max)) can be simplified to the following formulas ofEqs. (15)-(16):

W _(σ)(θ_(max))=½(L+{square root}(A ² +D ²))  (15)

θ_(max)=½a tan(D/A)  (16)

[0060] In the above formulas, the array L is defined as follows:

L=W _(σ()0)+W _(σ)(90)=I _(G) *D ₂(0)+I _(G) *D ₂(90)=I_(G) *[D₂(0)+D₂(90)]  (17)

[0061] $\begin{matrix}\begin{matrix}\quad & 0 & {- 1} & 0 & \quad & \quad \\{L =} & I_{G} & * & {- 1} & 4 & {- 1} \\\quad & 0 & {- 1} & 0 & \quad & \quad\end{matrix} & (18)\end{matrix}$

[0062] As known in the art, the array L is the result of the convolutionof I_(G) with a Laplacian operator. Furthermore, the array A in Eqs.(15) and (16) is defined as follows:

A=W _(σ)(0)−W _(σ)(90)=I _(G) *D ₂(0)−I _(G) *D ₂(90)=I _(G) *[D₂(0)−D₂(90)]  (19)

[0063] $\begin{matrix}\begin{matrix}\quad & 0 & 1 & 0 & \quad & \quad \\{A =} & I_{G} & * & {- 1} & 0 & {- 1} \\\quad & 0 & 1 & 0 & \quad & \quad\end{matrix} & (20)\end{matrix}$

[0064] Finally, the array D in Eqs. (15) and (16) is defined as follows:

D=W _(σ)(45)−W _(σ)(135)=I _(G) *D ₂(45)−I _(G) *D ₂(135)=I _(G) *[D₂(45)−D ₂(135)]  (21)

[0065] $\begin{matrix}\begin{matrix}\quad & 1 & 0 & {- 1} & \quad & \quad \\{D =} & I_{G} & * & 0 & 0 & 0 \\\quad & {- 1} & 0 & 1 & \quad & \quad\end{matrix} & (22)\end{matrix}$

[0066] Accordingly, at step 508 the convolution of Eq. (20) is performedon the filtered image array I_(G) that results from the previous step506 to produce the array A. At step 510, the convolution of Eq. (22) isperformed on the filtered image array I_(G) to produce the array D, andat step 512, the convolution of Eq. (18) is performed to produce thearray L. Since they are independent of each other, the steps 508-512 maybe performed in parallel or in any order. At step 514, the lineintensity function W_(σ)(θ_(max)) is formed directly from the arrays L,A, and D in accordance with Eq. (15). Subsequent to step 514, at step516 the line image array L(i, j) is formed from the line intensityfunction W_(σ)(θ_(max)) using known methods such as a simplethresholding process or a modified thresholding process based on ahistogram of the line intensity function W_(σ)(θ_(max)).

[0067] Finally, at step 518, the direction image array θ_(max)(i, j) isformed from the arrays D and A in accordance with Eq. (16).Advantageously, according to the preferred embodiment of FIG. 5, thestep 518 of computing the direction image array θ_(max)(i, j) and thesteps 514-516 of generating the line image array L(i, j) may beperformed independently of each other and in any order. Stated anotherway, according to the preferred embodiment of FIG. 5, it is notnecessary to actually compute the elements of the direction imageθ_(max)(i, j) in order to evaluate the line intensity estimator functionW₉₄ (θ_(max)) at any pixel. This is in contrast to the algorithmsdescribed in FIG. 2 and FIGS. 3 and 4, where it is first necessary tocompute the direction image θ_(max)(i, j) in order to be able toevaluate the line intensity estimator function W_(σ)(θ) at the maximumangle θ_(max).

[0068] It is readily apparent that in the preferred embodiment of FIG.5, steps 512, 514, and 516 may be omitted altogether if downstreammedical image processing algorithms only require knowledge of thedirection image array θ_(max)(i, j). Alternatively, the step 518 may beomitted altogether if downstream medical image processing algorithmsonly require knowledge of the line image array L(i, j). Thus,computational independence of the direction image array θ_(max)(i, j)and the line image array L(i, j) in the preferred embodiment of FIG. 5allows for increased computational efficiency when only one or the otherof the direction image array θ_(max)(i, j) and the line image array L(i,j) is required by downstream algorithms.

[0069] The preferred embodiment of FIG. 5 is even less computationallycomplex than the algorithm of FIG. 3 and 4. In particular, to generatethe filtered image array I_(G) there is required only approximatelyN_(k)·M·N multiplications and 2·N_(k)·M·N additions. To generate thearray A from the filtered image array I_(G), there is required 2·M·Nadditions and M·N subtractions. Likewise, to generate the array D fromthe filtered image array I_(G), there is required 2·M·N additions andM·N subtractions. Finally, to generate L from the filtered image arrayI_(G), there is required M·N bitwise left register shift of twopositions (corresponding to a multiplication by 4), followed by 4·M·Nsubtractions. Accordingly, to generate the arrays A, D, and L from thedigital mammogram image I, there is required only 2·N_(k)·M·Nmultiplications, 2·N_(k)M·N additions, 4·M·N additions, 4·M·Nsubtractions, and M·N bitwise register shifts.

[0070] For illustrative purposes in comparing the algorithms, let usagain assume the operational parameters assumed previously: thataddition, subtraction, and register-shifting operation take 10 clockcycles each; that multiplication takes 30 clock cycles; thatM×N=1000×1250; and that N_(k) is 11. As computed previously, theKarssemeijer algorithm would require 18.2 billion clock cycles tocompute the three line estimator functions W_(σ)(0), W_(σ)(60), andW_(σ)(120), while the algorithm of FIGS. 3 and 4 would require about 6.2billion clock cycles to generate the four line operator functionsW_(σ)(0), W_(σ)(45), W_(σ)(90), and W_(σ)(135), a significantcomputational advantage. However, using the results of the previousparagraph, the algorithm of FIG. 5 would require only(11)(1000)(1250)(30)+2(11)(1000)(1250)(10)+(4)(1000)(1250)(10)+(4)(1000)(1250)(10)+(1000)(1250)(10)=0.8billion clock cycles to produce the arrays A, D, and L. For thepreferred embodiment of FIG. 5, the reduction in computation becomeseven more dramatic as the scale of interest (reflected by the size ofthe kernel size N_(k)) grows larger, because the number of computationsonly increases linearly with N_(k). It is to be appreciated that theabove numerical example is a rough estimate and is for illustrativepurposes only to clarify the features and advantages of the presentinvention, and is not intended to limit the scope of the preferredembodiments.

[0071] Optionally, in the preferred embodiment of FIGS. 3-5, a pluralityof spatial scale values σ1, σ2, . . . , σn may be selected at step 302or 502. The remainder of the steps of the embodiments of FIGS. 3-5 arethen separately carried out for each of the spatial scale values σ1, σ2,. . . , σn. For a given pixel (i, j), the value of the direction imagearray θ_(max)(i, j) is selected to correspond to the largest value amongW_(σ1)(θ_(max1)), W_(σ2)(θ_(max2)), . . . , W_(σn)(θ_(maxn)). The lineimage array L(i, j) is formed by thresholding an array corresponding tolargest value among W_(σ1)(θ_(max1)), W_(σ2)(θ_(max2)), . . . ,W_(σn)(θ_(maxn)) at each pixel.

[0072] As another option, which may be used separately or in combinationwith the above option of using multiple spatial scale values, aplurality of filter kernel sizes N_(k1), N_(k2), . . . , N_(kn) ay beselected at step 302 or 502. The remainder of the steps of theembodiments of FIGS. 3-5 are then separately carried out for each of thefilter kernel sizes N_(k1), N_(k2), . . . , N_(kn). For a given pixel(i, j), the value of the direction image array θ_(max)(i, j) is selectedto correspond to the largest one of the different W_(σ)(θ_(max)) valuesyielded for the different values of filter kernel size N_(k). The lineimage array L(i, j) is formed by thresholding an array corresponding tolargest value among the different W_(σ)(θ_(max)) values yielded by thedifferent values of filter kernel size N_(k). By way of example and notby way of limitation, it has been found that with reference to thepreviously disclosed system for detecting lines in fibrous breast tissuein a 1000×1250 digital mammogram at 200 micron resolution, results aregood when the pair of combinations (N_(k)=11, σ=1.5) and (N_(k)=7,σ=0.9) are used.

[0073] The preferred embodiments disclosed in FIGS. 3-5 require acorrective algorithm to normalize the responses of certain portions ofthe algorithms associated with directional second order derivatives indiagonal directions. In particular, the responses of Eqs. (6b), (6d),and (22) require normalization because the filtered image is beingsampled at more widely displaced points, resulting in a response that istoo large by a constant factor. In the preferred algorithms that use aGaussian filter G at step 304 of FIG. 3 or steps 504-506 of FIG. 5, aconstant correction factor “p” is determined as shown in Eqs. (23)-(25):

p=SQRT{Σ(K _(A)(i,j))²/Σ(K _(D)(i,j))²}  (23)

[0074] $\begin{matrix}\begin{matrix}\quad & 0 & 1 & 0 & \quad & \quad \\{K_{A} =} & G & * & {- 1} & 0 & {- 1} \\\quad & 0 & 1 & 0 & \quad & \quad\end{matrix} & (24)\end{matrix}$

$\begin{matrix}\begin{matrix}\quad & 1 & 0 & {- 1} & \quad & \quad \\{K_{D} =} & G & * & 0 & 0 & 0 \\\quad & {- 1} & 0 & 1 & \quad & \quad\end{matrix} & (25)\end{matrix}$

[0075] In the general case where the digital mammogram image I isconvolved with a single-peaked filter F at step 304 of FIG. 3 or steps504-506 of FIG. 5, the constant correction actor p is determined byusing F instead of G in Eqs. (24) and (25).

[0076] Importantly, the constant correction factor p does not actuallyaffect the number of computations in the convolutions of Eqs. (6b),(6d), and (22), but rather is incorporated into later parts of thealgorithm. In particular, in the algorithm of FIG. 3, the constantcorrection factor p is incorporated by substituting, for each instanceof W_(σ)(45) and W_(σ)(135) in Eqs. (8) and (9), and step 308, thequantities pW_(σ)(45) and pW_(σ)(135), respectively. In the algorithm ofFIG. 5, the constant correction factor p is incorporated bysubstituting, for each instance of D in Eqs. (15) and (16), and steps514 and 518, the quantity pD. Accordingly, the computational efficiencyof the preferred embodiments is maintained in terms of the reducednumber and complexity of required convolutions.

[0077] A computational simplification in the implementation of theconstant correction factor p is found where the size of the spatialscale parameter 6 corresponds to a relatively large number of pixels,e.g. on the order of 11 pixels or greater. In this situation theconstant correction factor p approaches the value of ½, the samplingdistance going up by a factor of {square root}2 and the magnitude of thesecond derivative estimate going up by the square of the samplingdistance. In such case, multiplication by the constant correction factorp is achieved by a simple bitwise right register shift.

[0078] As disclosed above, a method and system for line detection inmedical images according to the preferred embodiments contains severaladvantages. The preferred embodiments share the homogeneity, isotropy,and other desirable scale-space properties associated with theKarssemeijer method. However, as described above, the preferredembodiments significantly reduce the number of required computations.Indeed, for one of the preferred embodiments, running time increasesonly linearly with the scale of interest, thus typically requiring anorder of magnitude fewer operations in order to produce equivalentresults. For applications in which processing time is a constraint, thismakes the use of higher resolution images in order to improve linedetection accuracy more practical.

[0079] While preferred embodiments of the invention have been described,these descriptions are merely illustrative and are not intended to limitthe present invention. For example, although the component kernels ofthe separable single-peaked filter function are described above asone-dimensional kernels, the selection of appropriate two-dimensionalkernels as component kernels of the single-peaked filter function canalso result in computational efficiencies, where one of the dimensionsis smaller than the initial size of the single-peaked filter function.As another example, although the embodiments of the invention describedabove were in the context of medical imaging systems, those skilled inthe art will recognize that the disclosed methods and structures arereadily adaptable for broader image processing applications. Examplesinclude the fields of optical sensing, robotics, vehicular guidance andcontrol systems, synthetic vision, or generally any system requiring thegeneration of line images or direction images from an input image.

What is claimed is:
 1. A method for detecting lines in a digital image,comprising the steps of: filtering said digital image to produce afiltered image array; convolving said filtered image array with aplurality of second order difference operators designed to extractsecond order directional derivative information from said filtered imagearray in a predetermined set of directions; processing informationresulting from said step of convolving to produce a line image; whereinsaid predetermined set of directions is selected to correspond to anaspect ratio of said second order difference operators.
 2. The method ofclaim 1, wherein said second order difference operators are squarekernels, and wherein said predetermined set of directions includes thedirections of 0, 45, 90, and 135 degrees.
 3. The method of claim 2,wherein said second order difference operators are 3×3 kernels.
 4. Themethod of claim 3, said step of filtering said digital image arraycomprising the steps of: selecting a single-peaked filter kernel; andconvolving said digital mammogram image with said single-peaked filterkernel.
 5. The method of claim 4, wherein said single-peaked filterkernel is a separable function comprising the convolution of a first onedimensional kernel and a second one dimensional kernel, and wherein saidstep of convolving said digital mammogram image with said single-peakedfilter kernel comprises the steps of convolving said digital mammogramimage with said first one dimensional kernel and said second onedimensional kernel.
 6. The method of claim 5, wherein said single-peakedfilter kernel is a Gaussian.
 7. The method of claim 6, wherein said stepof convolving said filtered image array comprises the steps of:convolving said filtered image array with 3×3 second order differenceoperators designed to extract second order derivative information alongthe 45 degree and 135 degree directions; and subsequent to said stepconvolving said filtered image array with 3×3 second order differenceoperators designed to extract second order derivative information alongthe 45 degree and 135 degree directions, multiplying the results of saidstep by a constant correction factor to accommodate for more widelyspaced sampling along the diagonals.
 8. A method for detecting lines ina digital image, comprising the steps of: selecting a spatial scaleparameter, said spatial scale parameter corresponding to a desired rangeof line widths for detection; convolving said digital image with a firstone dimensional kernel and a second one dimensional kernel to produce afiltered image array, said first one dimensional kernel and said secondone dimensional kernel each having a size related to said spatial scaleparameter; producing a line image based on second-order spatialderivatives of said filtered image array; wherein said line image isproduced from said digital image using a number of computations that issubstantially proportional to the spatial scale parameter such that, asthe spatial scale parameter is increased, said number of computationsincreases at a rate that is less than the rate of increase of the squareof the spatial scale parameter.
 9. The method of claim 8, said step ofproducing a line image based on second-order spatial derivatives of saidfiltered image array further comprising the steps of: convolving saidfiltered image array with a plurality of second order differenceoperators designed to extract second order directional derivativeinformation from said filtered image array in a predetermined set ofdirections; and processing information resulting from said step ofconvolving to produce a line image; wherein said predetermined set ofdirections includes directions along the diagonals of the digitalmammogram image.
 10. The method of claim 9, wherein said second orderdifference operators are 3×3 kernels.
 11. The method of claim 10,wherein said first one dimensional kernel and said second onedimensional kernel are single-peaked functions each having an odd numberof elements.
 12. The method of claim 11, wherein said first onedimensional kernel and said second one dimensional kernel are Gaussians.13. A method for detecting lines in a digital image, comprising thesteps of: selecting a spatial scale parameter, said spatial scaleparameter corresponding to a desired range of line widths for detection;convolving said digital image with a first one dimensional kernel and asecond one dimensional kernel to produce a filtered image array, saidfirst one dimensional kernel and said second one dimensional kernel eachhaving a size related to said spatial scale parameter; separatelyconvolving said filtered image array with a first, second, and thirdsecond order difference operator to produce a first, second, and thirdresulting array, respectively; computing a direction image arraycomprising, at each pixel, a first predetermined scalar function ofcorresponding pixel values in said first, second, and third resultingarrays; computing a line intensity function array comprising, at eachpixel, a second predetermined scalar function of corresponding pixelvalues in said first, second, and third resulting arrays; and computinga line image array using information in said line intensity functionarray.
 14. The method of claim 13, wherein said first, second, and thirdsecond order difference operators each comprise a 3×3 matrix.
 15. Themethod of claim 14, wherein said first second order difference operatorcomprises the difference between a horizontal second order differenceoperator and a vertical difference operator.
 16. The method of claim 15,wherein said second order difference operator comprises the differencebetween a first diagonal second order difference operator and a seconddiagonal second order difference operator.
 17. The method of claim 16,wherein said third second order difference operator is a Laplacian. 18.The method of claim 17, wherein said first predetermined scalar functioncomprises the arctangent of the quotient of said corresponding pixelvalue in said second resulting array divided by said corresponding pixelvalue in said first resulting array.
 19. The method of claim 18, whereinsaid second predetermined scalar function comprises the sum of two timesthe corresponding pixel value in said third resulting array plus thesquare root of the sum of the squares of the corresponding pixel valuein said first resulting array and the corresponding pixel value in saidsecond resulting array.
 20. The method of claim 19, wherein said step ofcomputing a line image array using information in said line intensityfunction array comprises the step of using a modified thresholdingprocess based on a histogram of said line intensity function.
 21. Acomputer-readable medium which can be used for directing an apparatus todetect lines in a digital image, comprising: means for directing saidapparatus to filter said image to produce a filtered array; means fordirecting said apparatus to convolve said filtered image array with aplurality of second order difference operators designed to extractsecond order directional derivative information from said filtered imagearray in a predetermined set of directions; means for directing saidapparatus to process information resulting from said step of convolvingto produce a line image; wherein said predetermined set of directions isselected to correspond to an aspect ratio of said second orderdifference operators.
 22. The computer-readable medium of claim 21,wherein said second order difference operators are square kernels, andwherein said predetermined set of directions includes the directions of0, 45, 90, and 135 degrees.
 23. The computer-readable medium of claim22, wherein said second order difference operators are 3×3 kernels. 24.The computer-readable medium of claim 23, said means for directing saidapparatus to filter said image to produce a filtered array furthercomprising means for directing said apparatus to convolve said digitalmammogram image with a single-peaked filter kernel.
 25. Thecomputer-readable medium of claim 23, said means for directing saidapparatus to filter said image to produce a filtered array furthercomprising means for directing said apparatus to convolve said digitalmammogram image with a separable single-peaked filter kernel bysuccessively convolving said digital image with a first one dimensionalcomponent kernel and a second one dimensional component kernel of saidseparable single-peaked filter kernel.
 26. The computer-readable mediumof claim 25, wherein said separable single-peaked filter kernel is aGaussian.
 27. An apparatus for detecting lines in digital images, saidapparatus comprising: a first memory for storing a digital image; afirst convolution device capable of convolving said digital image with afirst one dimensional kernel and a second one dimensional kernel toproduce a filtered image array, said first one dimensional kernel andsaid second one dimensional kernel each having a size related to thesize of lines being detected; a second convolution device capable ofseparately convolving said filtered image array with a first, a second,and a third second order difference operator to produce a first, second,and third resulting array, respectively; a first processing devicecapable of computing a direction image array comprising, at each pixel,a first predetermined scalar function of corresponding pixel values insaid first, second, and third resulting arrays; a second processingdevice capable of computing a line intensity function array comprising,at each pixel, a second predetermined scalar function of correspondingpixel values in said first, second, and third resulting arrays; and athird processing device capable of computing a line image array usinginformation in said line intensity function array.
 28. The method ofclaim 27, wherein said first, second, and third second order differenceoperators each comprise a 3×3 matrix.
 29. The method of claim 28,wherein said first second order difference operator comprises thedifference between a horizontal second order difference operator and avertical difference operator.
 30. The method of claim 29, wherein saidsecond second order difference operator comprises the difference betweena first diagonal second order difference operator and a second diagonalsecond order difference operator.
 31. The method of claim 30, whereinsaid third second order difference operator is a Laplacian.
 32. Themethod of claim 31, wherein said first predetermined scalar functioncomprises the arctangent of the quotient of said corresponding pixelvalue in said second resulting array divided by said corresponding pixelvalue in said first resulting array.
 33. The method of claim 32, whereinsaid second predetermined scalar function comprises the sum of two timesthe corresponding pixel value in said third resulting array plus thesquare root of the sum of the squares of the corresponding pixel valuein said first resulting array and the corresponding pixel value in saidsecond resulting array.